Problem: Solve for $x$ : $ 4|x - 9| + 7 = -4|x - 9| + 8 $
Solution: Add $ {4|x - 9|} $ to both sides: $ \begin{eqnarray} 4|x - 9| + 7 &=& -4|x - 9| + 8 \\ \\ { + 4|x - 9|} && { + 4|x - 9|} \\ \\ 8|x - 9| + 7 &=& 8 \end{eqnarray} $ Subtract ${7}$ from both sides: $ \begin{eqnarray} 8|x - 9| + 7 &=& 8 \\ \\ { - 7} &=& { - 7} \\ \\ 8|x - 9| &=& 1 \end{eqnarray} $ Divide both sides by ${8}$ $ \dfrac{8|x - 9|} {{8}} = \dfrac{1} {{8}} $ Simplify: $ |x - 9| = \dfrac{1}{8}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 9 = -\dfrac{1}{8} $ or $ x - 9 = \dfrac{1}{8} $ Solve for the solution where $x - 9$ is negative: $ x - 9 = -\dfrac{1}{8} $ Add ${9}$ to both sides: $ \begin{eqnarray} x - 9 &=& -\dfrac{1}{8} \\ \\ {+ 9} && {+ 9} \\ \\ x &=& -\dfrac{1}{8} + 9 \end{eqnarray} $ Change the ${ + 9}$ to an equivalent fraction with a denominator of $8$ $ x = - \dfrac{1}{8} {+ \dfrac{72}{8}} $ $ x = \dfrac{71}{8} $ Then calculate the solution where $x - 9$ is positive: $ x - 9 = \dfrac{1}{8} $ Add ${9}$ to both sides: $ \begin{eqnarray} x - 9 &=& \dfrac{1}{8} \\ \\ {+ 9} && {+ 9} \\ \\ x &=& \dfrac{1}{8} + 9 \end{eqnarray} $ Change the ${ + 9}$ to an equivalent fraction with a denominator of $8$ $ x = \dfrac{1}{8} {+ \dfrac{72}{8}} $ $ x = \dfrac{73}{8} $ Thus, the correct answer is $x = \dfrac{71}{8} $ or $x = \dfrac{73}{8} $.